Householder triangularization of a quasimatrix

A standard algorithm for computing the QR factorization of a matrix A is Householder triangularization. Here this idea is generalized to the situation in which A is a quasimatrix, that is, a “matrix” whose “columns” are functions defined on an interval [a,b]. Applications are mentioned to quasimatrix leastsquares fitting, singular value decomposition, and determination of ranks, norms, and condition numbers, and numerical illustrations are presented using the chebfun system

Quantum machine learning: a classical perspective

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde